Problem 34 - Algebra for College Students

Question

Suppose that the daily profit from an ice cream stand is given by the equation \(p=2 n-4\), where \(n\) represents the number of gallons of ice cream mix used in a day, and \(p\) represents the number of dollars of profit. Label the horizontal axis \(n\) and the vertical axis \(p\), and graph the equation \(p=2 n-4\) for nonnegative values of \(n\).

Step-by-Step Solution

Answer

Graph a line intersecting at (0, -4) and (2, 0) with slope 2 for nonnegative \(n\).

Step 1: Identify the Equation Type

The equation provided is a linear equation in slope-intercept form, \(p = 2n - 4\), where \(n\) is the independent variable (x-axis), and \(p\) is the dependent variable (y-axis).

Step 2: Determine the Slope and y-intercept

In the equation \(p = 2n - 4\), the slope \(m\) is 2, and the y-intercept \(b\) is -4. This means that for every additional gallon of ice cream mix used, the profit increases by $2, and the graph will cross the y-axis at -4.

Step 3: Calculate Key Points

To draw the graph, calculate key points where \(n\) is nonnegative. For example, when \(n = 0\), \(p = 2(0) - 4 = -4\), and when \(n = 2\), \(p = 2(2) - 4 = 0\). These points are (0, -4) and (2, 0).

Step 4: Plot the Points on the Graph

Plot the calculated points (0, -4) and (2, 0) on a graph with \(n\) on the horizontal axis and \(p\) on the vertical axis. Also, mark a few more points such as \(n=3, p=2\), to ensure the line fully captures more of the range.

Step 5: Draw the Line

Using a ruler, connect the plotted points with a straight line, extending it for all nonnegative values of \(n\). This line represents the profit equation for nonnegative gallons of ice cream mix.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting a straight line that showcases the relationship between two variables. In our exercise, the equation is given as \(p=2n-4\). This is a linear equation, which means its graph will result in a straight line. The equation allows us to understand how changes in one variable, in this case, the number of gallons of ice cream mix \(n\), will affect the other variable, the profit \(p\).
To graph the equation effectively, we need to plot several specific points that lie on the line and connect these using a ruler or straightedge. This visual representation can be particularly helpful for understanding increases or decreases in profit as more gallons of ice cream mix are used.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as \(y=mx+b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. In our specific equation \(p=2n-4\), \(m=2\) and \(b=-4\).
The slope, here being \(2\), tells us that for each additional gallon of ice cream mix used, the profit increases by $2. This means the slope is a measure of the 'steepness' or 'incline' of the line on the graph, showing how quickly or slowly the profit changes with respect to the number of gallons.
The y-intercept is the point where the line crosses the y-axis, which in this case is \(-4\). This intercept illustrates the profit when no gallons of ice cream mix are used at all.

Profit Calculation

Calculating profit entails using the linear equation to predict profit based on different values of \(n\), the number of gallons of ice cream mix used. For instance, if no gallons are used, the profit is calculated as \(p=2(0)-4=-4\), indicating a $4 loss. Conversely, if 2 gallons are used, the profit becomes \(p=2(2)-4=0\), reflecting a balanced or break-even position, with no profit or loss.
Such calculations are critical for understanding profit patterns and making business decisions based on potential sales or usage amounts. It shows not just the immediate gain or loss but also helps in strategic planning by forecasting future profits depending on resource usage.

Coordinate Plane Plotting

Plotting on a coordinate plane requires an understanding of both the x-axis and y-axis. In our example, the horizontal axis, labeled \(n\), represents the independent variable, number of gallons used. The vertical axis, labeled \(p\), represents the dependent variable, or the profit.
When plotting points such as \((0, -4)\) and \((2, 0)\), it's crucial to accurately place these on the plane. Each point is a solution of the equation that satisfies \(p=2n-4\).

• The origin (0,0) is a point of reference where the axes meet.
• The line that forms when you connect points with a ruler is the graphical representation of the equation.

Every additional plotted point extends the line, providing a clearer picture of the relationship between variables and helping visualize profit changes over various amounts of ice cream mix.